A characterization of integral input-to-state stability

The notion of input-to-state stability (ISS) is now recognized as a central concept in nonlinear systems analysis. It provides a nonlinear generalization of finite gains with respect to supremum norms and also of finite L/sup 2/ gains. It plays a central role in recursive design, coprime factorizations, controllers for nonminimum phase systems, and many other areas. In this paper, a newer notion, that of integral input-to-state stability (iISS), is studied. The notion of iISS generalizes the concept of finite gain when using an integral norm on inputs but supremum norms of states, in that sense generalizing the linear "H/sup 2/" theory. It allows one to quantify sensitivity even in the presence of certain forms of nonlinear resonance. We obtain several necessary and sufficient characterizations of the iISS property, expressed in terms of dissipation inequalities and other alternative and nontrivial characterizations.

[1]  Wolfgang Hahn,et al.  Stability of Motion , 1967 .

[2]  Eduardo Sontag Smooth stabilization implies coprime factorization , 1989, IEEE Transactions on Automatic Control.

[3]  Eduardo Sontag Further facts about input to state stabilization , 1990 .

[4]  J. Aplevich,et al.  Lecture Notes in Control and Information Sciences , 1979 .

[5]  Xiaoming Hu On state observers for nonlinear systems , 1991 .

[6]  Zhong-Ping Jiang,et al.  Small-gain theorem for ISS systems and applications , 1994, Math. Control. Signals Syst..

[7]  Eduardo Sontag,et al.  On characterizations of the input-to-state stability property , 1995 .

[8]  Wei-Min Lu A class of globally stabilizing controllers for nonlinear systems , 1995 .

[9]  Wei-Min Lu A state-space approach to parameterization of stabilizing controllers for nonlinear systems , 1995, IEEE Trans. Autom. Control..

[10]  Eduardo Sontag,et al.  New characterizations of input-to-state stability , 1996, IEEE Trans. Autom. Control..

[11]  A. Teel,et al.  Singular perturbations and input-to-state stability , 1996, IEEE Trans. Autom. Control..

[12]  A. Schaft L2-Gain and Passivity Techniques in Nonlinear Control. Lecture Notes in Control and Information Sciences 218 , 1996 .

[13]  Yuandan Lin,et al.  A Smooth Converse Lyapunov Theorem for Robust Stability , 1996 .

[14]  Yuan Wang,et al.  Stabilization in spite of matched unmodeled dynamics and an equivalent definition of input-to-state stability , 1996, Math. Control. Signals Syst..

[15]  A. Isidori Global almost disturbance decoupling with stability for non minimum-phase single-input single-output , 1996 .

[16]  Mrdjan Jankovic,et al.  Integrator forwarding: A new recursive nonlinear robust design , 1997, Autom..

[17]  Eduardo Sontag,et al.  A notion of input to output stability , 1997, 1997 European Control Conference (ECC).

[18]  Eduardo Sontag,et al.  Output-to-state stability and detectability of nonlinear systems , 1997 .

[19]  J. Tsinias Input to state stability properties of nonlinear systems and applications to bounded feedback stabilization using saturation , 1997 .

[20]  Eduardo Sontag,et al.  Asymptotic stability equals exponential stability, and ISS equals finite energy gain---if you twist your eyes , 1998, math/9812137.

[21]  A. S. MorseCenter Certainty Equivalence Implies Detectability , 1998 .

[22]  Eduardo Sontag Comments on integral variants of ISS , 1998 .

[23]  D. Liberzon ISS and integral-ISS disturbance attenuation with bounded controls , 1999, Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304).

[24]  Petar V. Kokotovic,et al.  Constructive nonlinear control: Progress in the 90''s , 1999 .

[25]  Riccardo Marino,et al.  Nonlinear output feedback tracking with almost disturbance decoupling , 1999, IEEE Trans. Autom. Control..

[26]  David Angeli,et al.  Input-to-state stability of PD-controlled robotic systems , 1999, Autom..

[27]  Joao P. Hespanha,et al.  Supervisory control of integral-input-to-state stabilizing controllers , 1999, 1999 European Control Conference (ECC).

[28]  Eduardo Sontag,et al.  Notions of input to output stability , 1999, Systems & Control Letters.

[29]  Stefano Battilotti,et al.  Robust stabilization of nonlinear systems with pointwise norm-bounded uncertainties: a control Lyapunov function approach , 1999, IEEE Trans. Autom. Control..